Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 431 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 0.797·3-s + 4-s + 1.40·5-s + 0.797·6-s − 7-s − 8-s − 2.36·9-s − 1.40·10-s − 4.26·11-s − 0.797·12-s + 0.799·13-s + 14-s − 1.12·15-s + 16-s − 3.62·17-s + 2.36·18-s − 6.70·19-s + 1.40·20-s + 0.797·21-s + 4.26·22-s − 4.59·23-s + 0.797·24-s − 3.01·25-s − 0.799·26-s + 4.27·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.460·3-s + 0.5·4-s + 0.629·5-s + 0.325·6-s − 0.377·7-s − 0.353·8-s − 0.788·9-s − 0.445·10-s − 1.28·11-s − 0.230·12-s + 0.221·13-s + 0.267·14-s − 0.289·15-s + 0.250·16-s − 0.880·17-s + 0.557·18-s − 1.53·19-s + 0.314·20-s + 0.173·21-s + 0.909·22-s − 0.957·23-s + 0.162·24-s − 0.603·25-s − 0.156·26-s + 0.822·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6034\)    =    \(2 \cdot 7 \cdot 431\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6034} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3324331737$
$L(\frac12)$  $\approx$  $0.3324331737$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;431\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;431\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
7 \( 1 + T \)
431 \( 1 - T \)
good3 \( 1 + 0.797T + 3T^{2} \)
5 \( 1 - 1.40T + 5T^{2} \)
11 \( 1 + 4.26T + 11T^{2} \)
13 \( 1 - 0.799T + 13T^{2} \)
17 \( 1 + 3.62T + 17T^{2} \)
19 \( 1 + 6.70T + 19T^{2} \)
23 \( 1 + 4.59T + 23T^{2} \)
29 \( 1 + 6.21T + 29T^{2} \)
31 \( 1 + 6.61T + 31T^{2} \)
37 \( 1 - 5.42T + 37T^{2} \)
41 \( 1 + 4.51T + 41T^{2} \)
43 \( 1 - 0.780T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 3.36T + 53T^{2} \)
59 \( 1 - 5.39T + 59T^{2} \)
61 \( 1 + 5.80T + 61T^{2} \)
67 \( 1 + 4.88T + 67T^{2} \)
71 \( 1 - 9.37T + 71T^{2} \)
73 \( 1 + 3.99T + 73T^{2} \)
79 \( 1 - 5.28T + 79T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 + 3.54T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.153859651098244914304940411452, −7.45413039407363120318884718510, −6.57798251400104952086764886424, −5.88894940862496182844881802794, −5.61438916838281589932066678521, −4.52452645648218822052881839167, −3.51965406088593362977605485305, −2.36893830307936706951492017626, −2.03357268878670688624207583909, −0.31686712646607045448801382303, 0.31686712646607045448801382303, 2.03357268878670688624207583909, 2.36893830307936706951492017626, 3.51965406088593362977605485305, 4.52452645648218822052881839167, 5.61438916838281589932066678521, 5.88894940862496182844881802794, 6.57798251400104952086764886424, 7.45413039407363120318884718510, 8.153859651098244914304940411452

Graph of the $Z$-function along the critical line